chaotic dynamical system
As Strogatz says in reference , “No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition”.
Chaos is the aperiodic long-term in a deterministic system that exhibits sensitive dependence on initial conditions.
Aperiodic long-term means that there are trajectories which do not settle down to fixed points, periodic orbits (http://planetmath.org/Orbit), or quasiperiodic as . For the purposes of this definition, a trajectory which approaches a limit of as should be considered to have a fixed point at .
Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast; i.e. (http://planetmath.org/Ie), the system has a positive Liapunov exponent.
Strogatz notes that he favors additional constraints on the aperiodic long-term , but leaves open (http://planetmath.org/OpenQuestion) what form they may take. He suggests two alternatives to fulfill this:
Requiring that there exists an open set of initial conditions having aperiodic trajectories, or
If one picks a random initial condition then there must be a nonzero chance of the associated trajectory being aperiodic.
0.1 Further reading
B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science: Similarities and Dissimilarities. http://pespmc1.vub.ac.be/Einmag_Abstr/BCodenotti.html
Steven H. Strogatz, ”Nonlinear Dynamics and Chaos”. Westview Press, 1994.
|Title||chaotic dynamical system|
|Date of creation||2013-03-22 13:05:26|
|Last modified on||2013-03-22 13:05:26|
|Last modified by||bshanks (153)|
|Synonym||deterministic chaotic system|